Coordinate ring of $V \subset \mathbb{A}^n$ variety

Question

Let $V \subset \mathbb{A}^n$ be a nonempty variety and $I(V)$ ideal of V in $K[x_1,...,x_2]$.

My question is straightforward: how can I see an element of the quotient $\frac{K[x_1,...,x_2]}{I(V)}$?

I'm not referring to $\Gamma(V)=\frac{K[x_1,...,x_2]}{I(V)}$ identified with the subring of $F(V,K)$ consisting of all polynomial functions on $V$. I would like to know how to look at an element of this quotient ring.

thanks.

Answer 1

For examples:

Let be $V=\{(x,y) \in \Bbb A^2_k; y-x^2=0 \}$ (parabola). we have $k[V]=k[x,y]/(y-x^2) \cong k[x,x^2]\cong k[x]$

Let be $V=\{(x,y) \in \Bbb A^2_k; y^2-x^3=0 \}$ . we have $k[V]=k[x,y]/(y^2-x^3) \cong k[x] \oplus k[x] \bar y$ where $\bar y$ is the reduise class of $y$.